Problem: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $15.5$ years; the standard deviation is $2.2$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living less than $11.1$ years.
$15.5$ $13.3$ $17.7$ $11.1$ $19.9$ $8.9$ $22.1$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $15.5$ years. We know the standard deviation is $2.2$ years, so one standard deviation below the mean is $13.3$ years and one standard deviation above the mean is $17.7$ years. Two standard deviations below the mean is $11.1$ years and two standard deviations above the mean is $19.9$ years. Three standard deviations below the mean is $8.9$ years and three standard deviations above the mean is $22.1$ years. We are interested in the probability of a seal living less than $11.1$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the seals will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the seals will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $11.1$ years and the other half $({2.5\%})$ will live longer than $19.9$ years. The probability of a particular seal living less than $11.1$ years is ${2.5\%}$.